Clothes in Wind: I like to have some wire in the edges of the clothing to make it flap in the wind. But not a complete web of wire everywhere, or it will bend every time you grab it, even if there is no wind in that scene.
I tried to think of something I've done with clothing flapping, the only one that is online is Death's cloak in The Seventh Skol, when he first appears. https://www.youtube.com/watch?v=dRt9ZIX4l5g&t=49s I had just one wire running around the edge of the cloak.
It is extra work. But sometimes, if you can't stop the clothing twitching about while you animate, making it a windy day with deliberate movement will hide the unintended movements, so it's actually easier to get a shot that looks good.
Falling and running, just like flying, require a rig to support the puppet in the air while there are no feet on the ground. Some puppets have mount points on the armature so you can attach a screw to the hip or chest block and mount it on a jointed rig. But if the puppet isn't too heavy, some armature wire will do well enough. Some of my smaller puppets just have 2 lengths of 3mm armature wire with sharp points as the rig, that I can stab into the foam, to support the puppet. There are more wires in the rig, but two that poke into the puppet. With only one wire, the puppet might rotate, so 2 keeps it steady. Whatever the rig is, you'll usually need to be able to use some software with layers so you can erase the rig afterwards. You need a clean shot of the background without the puppet, so you can put the animated puppet shot on a layer above that, and when you erase the wire, the matching background shows through. I use TV Paint Animation, but Photoshop can also do it, and I think After Effects as well.